Question: Simplify and expand the following expression: $ \dfrac{3}{2n - 10}+ \dfrac{2}{5n + 50}+ \dfrac{2}{n^2 + 5n - 50} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{3}{2n - 10} = \dfrac{3}{2(n - 5)}$ We can factor a $5$ out of denominator in the second term: $ \dfrac{2}{5n + 50} = \dfrac{2}{5(n + 10)}$ We can factor the quadratic in the third term: $ \dfrac{2}{n^2 + 5n - 50} = \dfrac{2}{(n - 5)(n + 10)}$ Now we have: $ \dfrac{3}{2(n - 5)}+ \dfrac{2}{5(n + 10)}+ \dfrac{2}{(n - 5)(n + 10)} $ The least common multiple of the denominators is: $ 10(n - 5)(n + 10)$ In order to get the first term over $10(n - 5)(n + 10)$ , multiply by $\dfrac{5(n + 10)}{5(n + 10)}$ $ \dfrac{3}{2(n - 5)} \times \dfrac{5(n + 10)}{5(n + 10)} = \dfrac{15(n + 10)}{10(n - 5)(n + 10)} $ In order to get the second term over $10(n - 5)(n + 10)$ , multiply by $\dfrac{2(n - 5)}{2(n - 5)}$ $ \dfrac{2}{5(n + 10)} \times \dfrac{2(n - 5)}{2(n - 5)} = \dfrac{4(n - 5)}{10(n - 5)(n + 10)} $ In order to get the third term over $10(n - 5)(n + 10)$ , multiply by $\dfrac{10}{10}$ $ \dfrac{2}{(n - 5)(n + 10)} \times \dfrac{10}{10} = \dfrac{20}{10(n - 5)(n + 10)} $ Now we have: $ \dfrac{15(n + 10)}{10(n - 5)(n + 10)} + \dfrac{4(n - 5)}{10(n - 5)(n + 10)} + \dfrac{20}{10(n - 5)(n + 10)} $ $ = \dfrac{ 15(n + 10) + 4(n - 5) + 20} {10(n - 5)(n + 10)} $ Expand: $ = \dfrac{15n + 150 + 4n - 20 + 20}{10n^2 + 50n - 500} $ $ = \dfrac{19n + 150}{10n^2 + 50n - 500}$